Facts would be hard to find. I propose a study in which thousands of children are studied to determine how many hours of violent video games they play per week. They would be divided into quintiles based on this number. They would then be followed for 50 or 60 years to determine what percentage of the children in each quintile turn out to be mass murderers, or even just more violent than average. Of course, you'd have to find ways to control for other violent influences (movies, TV, real life, etc.) they are exposed to. You'd also have to find a way to keep them from making significant increases or decreases throughout their life, or at least track the changes (maybe outcome in only affected by violent video games during a particular critical developmental window).

With my kids, I tried to discourage the games that seemed extreme to me, and didn't worry about the others. So far, no mass murderers in the family.

That's not a good study. You need to select children randomly at birth and either force them to play or to not play such games. Otherwise you have a sample bias where violent kids are choosing to play violent games.

Also "violent" is vague. I think cartoonish violence has a different emotional impact.